Abstract : The object of the work presented here is to provide improved conceptual and computational tools for handling the class membership problem: to decide whether an observation consisting of numerical values assigned to each of N parameters is or is not due to some phenomenon known descriptively in terms of a sufficiently large set of such observations, or to which of several phenomena the observation should be assigned. To each data vector in N space, representing a single observation, are annexed the elements of higher order moment tensors constructed from it to form an 'extended vector.' From the extended vectors associated with a cluster of data points is constructed an 'extended covariance matrix' which serves to define distance from the cluster and in a fashion which reflects the shape of the given cluster. Distance of a new observation (data point) from a given cluster, as measured in this metric, provides a means for deciding whether the observation belongs to the cluster. A pattern, consisting of a large number of points in N space, is represented by its 'extended covariance matrix.' The latter can be regarded as a vector in 'pattern space' and hence amenable to the same considerations employed in the class assignment problem. (Author)